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Positive periodic solutions of delayed differential equations. (English) Zbl 1248.34104

Summary: In this paper, we are concerned with the existence, multiplicity and nonexistence of positive ?-periodic solutions of the following equation \[ u''(t)+ a(t,u) u(t)=\lambda b(t) f(u(t-\tau(t))),\quad t\in\mathbb{R}, \] where \(a(\cdot,\cdot)\in C(\mathbb{R}\times\mathbb{R}, \mathbb{R}^+)\) is a \(\omega\)-periodc function with respect to the first variable, \(b(\cdot)\in C(\mathbb{R},[0,\infty))\), \(\tau(\cdot)\in C(\mathbb{R}, \mathbb{R})\) are \(\omega\)-periodic functions, \(f\in C([0,\infty), [0,\infty))\) and \(f(s)> 0\) for \(s> 0\), \(\lambda> 0\) is a parameter. The proof of our main result is based upon fixed point index theory.

MSC:

34K13 Periodic solutions to functional-differential equations
47N20 Applications of operator theory to differential and integral equations
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